Problem: Let $a,$ $b,$ and $c$ be nonzero real numbers, and let
\[x = \frac{b}{c} + \frac{c}{b}, \quad y = \frac{a}{c} + \frac{c}{a}, \quad z = \frac{a}{b} + \frac{b}{a}.\]Simplify $x^2 + y^2 + z^2 - xyz.$
Answer: Substituting and expanding, we get
\begin{align*}
x^2 + y^2 + z^2 - xyz &= \left( \frac{b}{c} + \frac{c}{b} \right)^2 + \left( \frac{a}{c} + \frac{c}{a} \right)^2 + \left( \frac{a}{b} + \frac{b}{a} \right)^2 - \left( \frac{b}{c} + \frac{c}{b} \right) \left( \frac{a}{c} + \frac{c}{a} \right) \left( \frac{a}{b} + \frac{b}{a} \right) \\
&= \frac{b^2}{c^2} + 2 + \frac{c^2}{b^2} + \frac{a^2}{c^2} + 2 + \frac{c^2}{a^2} + \frac{a^2}{b^2} + 2 + \frac{b^2}{a^2} - \left( \frac{a^2}{c^2} + \frac{b^2}{c^2} + 1 + \frac{b^2}{a^2} + \frac{a^2}{b^2} + 1 + \frac{c^2}{b^2} + \frac{c^2}{a^2} \right) \\
&= \boxed{4}.
\end{align*}